The context is that I am reading the proof that Lebesgue measure is rotation invariant Let X be a k-dimensional euclidean space. T is a linear map and its range is a subspace Y of lower dimension. I want to prove that m(Y) = 0 where m is the lebesgue measure in X. How to prove this? Consider a special case k = 2.Consider the subspace which is a linear combination of (1,1) (Which is a line 45 degrees from the x-axis). Call the subspace as Y. How can we prove that m(Y) = 0 without rotating the Y. Also, is there a simple( and rigorous!) proof that rotation is a Linear Transformation in k-dimensional euclidean space?